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Suppose I receive a list of 1 million coinflips, and I want to know how likely it is that the list was randomly generated.

My first thought would be to count the number of heads and tails, which should be evenly distributed (around 500.000). But suppose the distribution looks normal, its still possible the list contains patterns or repititions. For example, the first half of the list may be the heads, and the last half the tails. In real random data, that would be highly unlikely.

So how do you calculate the 'randomness' of this list?

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possible duplicate of What tests can I do to ensure my PRNG is working correctly? –  otus Oct 16 at 14:12
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Short answer: You can't test if it's good. You can only check if it's bad in specific ways. –  CodesInChaos Oct 16 at 14:21
    
Maurer's universal statistical test (see A. J. Menezes et al., Handbook of Applied Cryptography, sec.5.4.5 (available online)) may be of some interest to you. –  Mok-Kong Shen Oct 16 at 19:05
    
See @e-sushi 's answer in the duplicate question for several good tests. –  John Deters Oct 17 at 3:07
    
@otus I disagree about your proposed duplicate: testing the randomess of a CSPRNG output is completely different from testing the randomness of an entropy source. For one thing, the former is just about useless, the latter is useful. –  Gilles Oct 17 at 9:42

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You can never actually prove that it was generated randomly or pseudorandomly. You can only prove with high probability that it wasn't. Calculating the number of heads and tails is one way. Another is calculating runs of consecutive heads or tails. There is a suite of statistical tests from NIST in their FIPS 140-2 document which is a good place to start.

Having said that, for cryptographic purposes you really need to be sure that you are using a secure random number generator and there aren't any tests you can apply to the data itself to sufficiently guarantee that it is secure enough.

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Can you never prove pseudorandomness or do we just not know how? Existance of one-way functions would imply existence of pseudorandom generators right? It would also imply $P\neq NP$. –  mikeazo Oct 17 at 18:24
    
Given an algorithm, you can prove that it's output is pseudorandom based on some computational assumption, but that is as close as you can get. For instance, Blum-Blum-Shub is a PRNG that is pseudorandom if factoring is a hard problem. As you say, PRNG implies $P \neq NP$, which is not known. –  Travis Mayberry Oct 17 at 19:37

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